/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 6053 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   and(false, false) -> false
   and(true, false) -> false
   and(false, true) -> false
   and(true, true) -> true
   eq(nil, nil) -> true
   eq(cons(T, L), nil) -> false
   eq(nil, cons(T, L)) -> false
   eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
   eq(var(L), var(Lp)) -> eq(L, Lp)
   eq(var(L), apply(T, S)) -> false
   eq(var(L), lambda(X, T)) -> false
   eq(apply(T, S), var(L)) -> false
   eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
   eq(apply(T, S), lambda(X, Tp)) -> false
   eq(lambda(X, T), var(L)) -> false
   eq(lambda(X, T), apply(Tp, Sp)) -> false
   eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
   if(true, var(K), var(L)) -> var(K)
   if(false, var(K), var(L)) -> var(L)
   ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
   ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
   ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T)))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   and(false, false) -> false
   and(true, false) -> false
   and(false, true) -> false
   and(true, true) -> true
   eq(nil, nil) -> true
   eq(cons(T, L), nil) -> false
   eq(nil, cons(T, L)) -> false
   eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
   eq(var(L), var(Lp)) -> eq(L, Lp)
   eq(var(L), apply(T, S)) -> false
   eq(var(L), lambda(X, T)) -> false
   eq(apply(T, S), var(L)) -> false
   eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
   eq(apply(T, S), lambda(X, Tp)) -> false
   eq(lambda(X, T), var(L)) -> false
   eq(lambda(X, T), apply(Tp, Sp)) -> false
   eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
   if(true, var(K), var(L)) -> var(K)
   if(false, var(K), var(L)) -> var(L)
   ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
   ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
   ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T)))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
Innermost TRS:
Rules:
and(false, false) -> false
and(true, false) -> false
and(false, true) -> false
and(true, true) -> true
eq(nil, nil) -> true
eq(cons(T, L), nil) -> false
eq(nil, cons(T, L)) -> false
eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
eq(var(L), var(Lp)) -> eq(L, Lp)
eq(var(L), apply(T, S)) -> false
eq(var(L), lambda(X, T)) -> false
eq(apply(T, S), var(L)) -> false
eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
eq(apply(T, S), lambda(X, Tp)) -> false
eq(lambda(X, T), var(L)) -> false
eq(lambda(X, T), apply(Tp, Sp)) -> false
eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
if(true, var(K), var(L)) -> var(K)
if(false, var(K), var(L)) -> var(L)
ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T)))

Types:
and :: false:true -> false:true -> false:true
false :: false:true
true :: false:true
eq :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> false:true
nil :: nil:cons:var:apply:lambda
cons :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
var :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
apply :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
lambda :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
if :: false:true -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
ren :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
hole_false:true1_0 :: false:true
hole_nil:cons:var:apply:lambda2_0 :: nil:cons:var:apply:lambda
gen_nil:cons:var:apply:lambda3_0 :: Nat -> nil:cons:var:apply:lambda

----------------------------------------

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
eq, ren

They will be analysed ascendingly in the following order:
eq < ren

----------------------------------------

(6)
Obligation:
Innermost TRS:
Rules:
and(false, false) -> false
and(true, false) -> false
and(false, true) -> false
and(true, true) -> true
eq(nil, nil) -> true
eq(cons(T, L), nil) -> false
eq(nil, cons(T, L)) -> false
eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
eq(var(L), var(Lp)) -> eq(L, Lp)
eq(var(L), apply(T, S)) -> false
eq(var(L), lambda(X, T)) -> false
eq(apply(T, S), var(L)) -> false
eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
eq(apply(T, S), lambda(X, Tp)) -> false
eq(lambda(X, T), var(L)) -> false
eq(lambda(X, T), apply(Tp, Sp)) -> false
eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
if(true, var(K), var(L)) -> var(K)
if(false, var(K), var(L)) -> var(L)
ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T)))

Types:
and :: false:true -> false:true -> false:true
false :: false:true
true :: false:true
eq :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> false:true
nil :: nil:cons:var:apply:lambda
cons :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
var :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
apply :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
lambda :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
if :: false:true -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
ren :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
hole_false:true1_0 :: false:true
hole_nil:cons:var:apply:lambda2_0 :: nil:cons:var:apply:lambda
gen_nil:cons:var:apply:lambda3_0 :: Nat -> nil:cons:var:apply:lambda


Generator Equations:
gen_nil:cons:var:apply:lambda3_0(0) <=> nil
gen_nil:cons:var:apply:lambda3_0(+(x, 1)) <=> cons(nil, gen_nil:cons:var:apply:lambda3_0(x))


The following defined symbols remain to be analysed:
eq, ren

They will be analysed ascendingly in the following order:
eq < ren

----------------------------------------

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eq(gen_nil:cons:var:apply:lambda3_0(n5_0), gen_nil:cons:var:apply:lambda3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
eq(gen_nil:cons:var:apply:lambda3_0(0), gen_nil:cons:var:apply:lambda3_0(0)) ->_R^Omega(1)
true

Induction Step:
eq(gen_nil:cons:var:apply:lambda3_0(+(n5_0, 1)), gen_nil:cons:var:apply:lambda3_0(+(n5_0, 1))) ->_R^Omega(1)
and(eq(nil, nil), eq(gen_nil:cons:var:apply:lambda3_0(n5_0), gen_nil:cons:var:apply:lambda3_0(n5_0))) ->_R^Omega(1)
and(true, eq(gen_nil:cons:var:apply:lambda3_0(n5_0), gen_nil:cons:var:apply:lambda3_0(n5_0))) ->_IH
and(true, true) ->_R^Omega(1)
true

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(8)
Complex Obligation (BEST)

----------------------------------------

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
and(false, false) -> false
and(true, false) -> false
and(false, true) -> false
and(true, true) -> true
eq(nil, nil) -> true
eq(cons(T, L), nil) -> false
eq(nil, cons(T, L)) -> false
eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
eq(var(L), var(Lp)) -> eq(L, Lp)
eq(var(L), apply(T, S)) -> false
eq(var(L), lambda(X, T)) -> false
eq(apply(T, S), var(L)) -> false
eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
eq(apply(T, S), lambda(X, Tp)) -> false
eq(lambda(X, T), var(L)) -> false
eq(lambda(X, T), apply(Tp, Sp)) -> false
eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
if(true, var(K), var(L)) -> var(K)
if(false, var(K), var(L)) -> var(L)
ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T)))

Types:
and :: false:true -> false:true -> false:true
false :: false:true
true :: false:true
eq :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> false:true
nil :: nil:cons:var:apply:lambda
cons :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
var :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
apply :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
lambda :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
if :: false:true -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
ren :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
hole_false:true1_0 :: false:true
hole_nil:cons:var:apply:lambda2_0 :: nil:cons:var:apply:lambda
gen_nil:cons:var:apply:lambda3_0 :: Nat -> nil:cons:var:apply:lambda


Generator Equations:
gen_nil:cons:var:apply:lambda3_0(0) <=> nil
gen_nil:cons:var:apply:lambda3_0(+(x, 1)) <=> cons(nil, gen_nil:cons:var:apply:lambda3_0(x))


The following defined symbols remain to be analysed:
eq, ren

They will be analysed ascendingly in the following order:
eq < ren

----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
Obligation:
Innermost TRS:
Rules:
and(false, false) -> false
and(true, false) -> false
and(false, true) -> false
and(true, true) -> true
eq(nil, nil) -> true
eq(cons(T, L), nil) -> false
eq(nil, cons(T, L)) -> false
eq(cons(T, L), cons(Tp, Lp)) -> and(eq(T, Tp), eq(L, Lp))
eq(var(L), var(Lp)) -> eq(L, Lp)
eq(var(L), apply(T, S)) -> false
eq(var(L), lambda(X, T)) -> false
eq(apply(T, S), var(L)) -> false
eq(apply(T, S), apply(Tp, Sp)) -> and(eq(T, Tp), eq(S, Sp))
eq(apply(T, S), lambda(X, Tp)) -> false
eq(lambda(X, T), var(L)) -> false
eq(lambda(X, T), apply(Tp, Sp)) -> false
eq(lambda(X, T), lambda(Xp, Tp)) -> and(eq(T, Tp), eq(X, Xp))
if(true, var(K), var(L)) -> var(K)
if(false, var(K), var(L)) -> var(L)
ren(var(L), var(K), var(Lp)) -> if(eq(L, Lp), var(K), var(Lp))
ren(X, Y, apply(T, S)) -> apply(ren(X, Y, T), ren(X, Y, S))
ren(X, Y, lambda(Z, T)) -> lambda(var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), ren(X, Y, ren(Z, var(cons(X, cons(Y, cons(lambda(Z, T), nil)))), T)))

Types:
and :: false:true -> false:true -> false:true
false :: false:true
true :: false:true
eq :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> false:true
nil :: nil:cons:var:apply:lambda
cons :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
var :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
apply :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
lambda :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
if :: false:true -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
ren :: nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda -> nil:cons:var:apply:lambda
hole_false:true1_0 :: false:true
hole_nil:cons:var:apply:lambda2_0 :: nil:cons:var:apply:lambda
gen_nil:cons:var:apply:lambda3_0 :: Nat -> nil:cons:var:apply:lambda


Lemmas:
eq(gen_nil:cons:var:apply:lambda3_0(n5_0), gen_nil:cons:var:apply:lambda3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_nil:cons:var:apply:lambda3_0(0) <=> nil
gen_nil:cons:var:apply:lambda3_0(+(x, 1)) <=> cons(nil, gen_nil:cons:var:apply:lambda3_0(x))


The following defined symbols remain to be analysed:
ren